Problem 27 - College Algebra

Question

For the following exercises, suppose \(\log _{5}(6)=a\) and \(\log _{5}(11)=b\). Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b\). Show the steps for solving. \(\log _{11}(5)\)

Step-by-Step Solution

Answer

\(\log_{11}(5) = \frac{1}{b}\)

Step 1: Use the Change of Base Formula

The change of base formula for logarithms allows us to rewrite a logarithm in terms of other logarithms. For our problem, we'll use the change of base formula:\[\log_{11}(5) = \frac{\log_{5}(5)}{\log_{5}(11)}\]

Step 2: Simplify the Numerator

The numerator \(\log_{5}(5)\) can be simplified using the property \(\log_{b}(b) = 1\). Therefore, \(\log_{5}(5) = 1\).

Step 3: Substitute Known Values

Now substitute the known value of \(\log_{5}(11)\). We know \(\log_{5}(11) = b\), so our expression becomes:\[\log_{11}(5) = \frac{1}{b}\]

Step 4: Final Expression

Thus, the logarithm \(\log_{11}(5)\) is rewritten in terms of \(b\):\[\log_{11}(5) = \frac{1}{b}\]

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool in logarithms that helps transform expressions between different bases. This is useful for calculations or simplifications when the base you want to work with is not directly accessible or familiar. The formula states that for any logarithm \(\log_{c}(a)\), you can express it in terms of a new base \(b\) as follows:

• \(\log_{c}(a) = \frac{\log_{b}(a)}{\log_{b}(c)}\)

This formula allows for the transformation of a logarithm from one base to another by employing the logarithms of the desired base.
To better understand this, consider the example \(\log_{11}(5)\). To convert this into base 5, you would express it as \(\log_{11}(5) = \frac{\log_{5}(5)}{\log_{5}(11)}\). From here, we can leverage any known values or simplifications from base 5.

Properties of Logarithms

Understanding the properties of logarithms is crucial because they simplify operations and make it easier to solve complicated logarithmic functions. Here's a quick overview of essential properties:

• The product property: \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\)
• The quotient property: \(\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)\)
• The power property: \(\log_{b}(x^n) = n \cdot \log_{b}(x)\)
• The identity property: \(\log_{b}(b) = 1\)

For example, when dealing with \(\log_{5}(5)\), the identity property simplifies it to 1. This is because any number raised to the power of one equals itself. These properties are handy when manipulating or rearranging logarithmic expressions without changing their value.

Simplifying Expressions

Simplifying logarithmic expressions often involves combining the change of base formula and the properties of logarithms to transform the expression into a simpler form. This process can reduce complexity and facilitate understanding or computation.
Consider the expression \(\log_{11}(5)\). By applying the change of base formula, the expression becomes \(\frac{\log_{5}(5)}{\log_{5}(11)}\). Using the identity property, we simplify \(\log_{5}(5)\) to 1.
Thus, the expression reduces to \(\frac{1}{\log_{5}(11)}\), where we substitute the given value of \(\log_{5}(11) = b\). This simplifies our expression to \(\frac{1}{b}\).
Simplified expressions like \(\frac{1}{b}\) are easier to interpret or use in further calculations or comparisons, highlighting the importance of these techniques in algebra and calculus.